Given:
1. The arc length [tex]\( L \)[/tex] is [tex]\( 32\pi \)[/tex] centimeters.
2. The central angle [tex]\( \theta \)[/tex] is [tex]\( \frac{8}{9}\pi \)[/tex] radians.
We can use the formula that relates the arc length of a circle, its radius, and the central angle:
[tex]\[
L = r \theta
\][/tex]
To find the radius [tex]\( r \)[/tex], we rearrange the formula:
[tex]\[
r = \frac{L}{\theta}
\][/tex]
Substituting the given values:
[tex]\[
r = \frac{32\pi}{\frac{8}{9}\pi}
\][/tex]
Simplify the expression:
[tex]\[
r = \frac{32\pi}{\frac{8\pi}{9}} = \frac{32\pi \cdot 9}{8\pi} = \frac{32 \cdot 9}{8} = \frac{288}{8} = 36
\][/tex]
Therefore, the radius of the circle is [tex]\( 36 \)[/tex] centimeters.
